Graph classes with given 3-connected components: asymptotic counting and critical phenomena

Fix a family T of 3-connected graphs, and let G be the class of graphs whose 3-connected components are the graphs in T. We present a general framework for analyzing such graph classes based on singularity analysis of generating functions. This generalizes previously studied cases such as planar graphs and series-parallel graphs. We provide a general theorem for the asymptotic number of graphs in G, based on the singularities of the exponential generating function associated to T. For some of the classes under study we show the presence of critical phenomena as the edge density in the class varies.